On SpyreStudios.com there's a post showing you how to connect to the Instagr.am API and pull data back with a bit of simple PHP.
The process of connecting into API data via OAuth is tricky. Some of the more notable social networks which have adopted this technology include Foursquare, Twitter, and Facebook. But Instagram is a very popular mobile app which has grown tremendously fast. Their recent API v1.0 release has some really neat features where you can pull popular photos, recent users, follower lists, and a whole bunch more.
He points to the Instagr.am PHP API scripts that the service already provides and shows how to set up an application on your Instagr.am account. This application then has OAuth keys you can use to connect to the service (via the PHP API scripts) and pull back basic user information - profile picture, number of photos, followers, etc. There's also a live demo you can test out.